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Theorem dnnumch2 41401
Description: Define an enumeration (weak dominance version) of a set from a choice function. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypotheses
Ref Expression
dnnumch.f 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
dnnumch.a (𝜑𝐴𝑉)
dnnumch.g (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
Assertion
Ref Expression
dnnumch2 (𝜑𝐴 ⊆ ran 𝐹)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐺,𝑧   𝑦,𝐴,𝑧
Allowed substitution hints:   𝜑(𝑦,𝑧)   𝐹(𝑧)   𝑉(𝑦,𝑧)

Proof of Theorem dnnumch2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dnnumch.f . . 3 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧))))
2 dnnumch.a . . 3 (𝜑𝐴𝑉)
3 dnnumch.g . . 3 (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺𝑦) ∈ 𝑦))
41, 2, 3dnnumch1 41400 . 2 (𝜑 → ∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴)
5 f1ofo 6796 . . . . . 6 ((𝐹𝑥):𝑥1-1-onto𝐴 → (𝐹𝑥):𝑥onto𝐴)
6 forn 6764 . . . . . 6 ((𝐹𝑥):𝑥onto𝐴 → ran (𝐹𝑥) = 𝐴)
75, 6syl 17 . . . . 5 ((𝐹𝑥):𝑥1-1-onto𝐴 → ran (𝐹𝑥) = 𝐴)
8 resss 5967 . . . . . 6 (𝐹𝑥) ⊆ 𝐹
9 rnss 5899 . . . . . 6 ((𝐹𝑥) ⊆ 𝐹 → ran (𝐹𝑥) ⊆ ran 𝐹)
108, 9mp1i 13 . . . . 5 ((𝐹𝑥):𝑥1-1-onto𝐴 → ran (𝐹𝑥) ⊆ ran 𝐹)
117, 10eqsstrrd 3988 . . . 4 ((𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹)
1211a1i 11 . . 3 (𝜑 → ((𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹))
1312rexlimdvw 3158 . 2 (𝜑 → (∃𝑥 ∈ On (𝐹𝑥):𝑥1-1-onto𝐴𝐴 ⊆ ran 𝐹))
144, 13mpd 15 1 (𝜑𝐴 ⊆ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  Vcvv 3448  cdif 3912  wss 3915  c0 4287  𝒫 cpw 4565  cmpt 5193  ran crn 5639  cres 5640  Oncon0 6322  ontowfo 6499  1-1-ontowf1o 6500  cfv 6501  recscrecs 8321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322
This theorem is referenced by:  dnnumch3lem  41402  dnnumch3  41403
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